A New Iterative Numerical Continuation Technique for Approximating the Solutions of Scalar Nonlinear Equations

<div>Evolution of the approximate solutions (<svg height="16.3113pt" id="M323" style="vertical-align:-4.053101pt" version="1.1" viewbox="-0.0498162 -12.2582 52.0902 16.3113" width="52.0902pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M715 650H476L470 622C550 619 562 610 562 570C562 536 553 481 535 384L523 321C485 121 408 37 283 37C178 37 99 109 134 297L178 532C193 612 199 616 278 622L284 650H29L23 622C105 616 109 609 94 532L48 291C23 160 51 83 98 40C143 -1 202 -16 260 -16C321 -16 387 3 439 49C509 111 542 204 563 312L577 384C595 479 608 533 619 569C629 603 637 620 709 622L715 650Z" id="g113-86"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="727" vert-adv-y="727"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,9.454,-5.741)"><path d="M485 418C485 434 469 451 441 451C389 451 312 387 255 335C221 304 194 278 159 242H157L259 678C264 698 264 710 255 710C242 710 183 681 96 673L90 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descent="-2856" horiz-adv-x="509" vert-adv-y="509"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,36.714,3.784)"><path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z" id="g50-111"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="512" vert-adv-y="512"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,41.374,3.784)"><path d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z" id="g54-36"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,46.934,3.784)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg>) obtained with the new iterative numerical continuation algorithm (blue solid line with circles; see (<a href="https://static.hindawi.com/articles/ijem/volume-2017/2783682/figures/#EEq27">26</a>)–(<a href="https://static.hindawi.com/articles/ijem/volume-2017/2783682/figures/#EEq31">30</a>) with <svg height="11.8174pt" id="M324" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -8.6359 39.2555 11.8174" width="39.2555pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M377 665C166 665 44 494 44 317C44 126 183 -15 363 -15C547 -15 695 118 695 333C695 533 546 665 377 665ZM355 630C491 630 593 515 593 304C593 112 496 22 385 22C245 22 146 157 146 345S244 630 355 630ZM502 225V442H471C464 375 459 369 411 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International Journal of Engineering Mathematics

fig5

Figure 5

Figure 5: A New Iterative Numerical Continuation Technique for Approximating the Solutions of Scalar Nonlinear Equations 